what do you mean by Measures of Central Tendency? define Arithmetic Mean, Mediam and Mode.
The central tendency of a variable means a typical value around which other values tend to concentrate; hence this value representing the central tendency of the series is called measures of central tendency or average.
According to Clark, “Average is an attempt to find one single figure to describe whole of figures.”
Arithmetic Mean (X) : The most popular and widely used measure of representing the entire data by one value is known as arithmetic mean. Its value is obtained by adding together all the items and by dividing this total by the number of items.
Arithmetic mean may be of two types :
- Simple Arithmetic mean
- Weighted arithmetic mean
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Individual Series |
Discrete & continuous series |
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Direct Method : X = ∑X N Where – |
Direct Method : X = ∑f X N Where – |
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X à Values N àNo. of Items |
fX àValues X frequencies N àTotal of frequencies |
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Shortcut Method : X = A + ∑dx
N Where – A àAssumed Mean dx à X - A |
Shortcut Method :
X = A + ∑f dx
N |
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Step-Deviation Method : Not applicable |
Step-Deviation Method :
X = A + ∑f dx ‘ X i
N Where – dx ‘ àX – A i i à Class interval |
where N –> No. of items
b) Discrete Series :
Arrange the variables in ascending or descending order.
Calculate cumulative frequencies.
Apply formula M = th item; N = Total of frequency.
The value (X) corresponding to this in the cumulative frequency will be the median.
b) Continuous Series :
Arrange the variables in ascending or descending order.
Calculate cumulative frequencies. Determine median class by using (N/2). Apply formula –
M = l1 + i (m – c)
f
l1 à lower limit of median group i à class interval
m àN/2
c àcumulative frequency preceding the median group. f à frequency of median group
Mode (Z) : Mode is the value that appears most frequently in a series i.e. it is the value of the item around which frequencies are most densely concentrated.
Calculation of Mode :
- Individual Series :
- By inspection – value repeated
- By converting individual series into discrete
By empirical relationship between the averages -
Z = 3M – 2X
b) Discrete Series :
- Arrange the variables in ascending or descending order.
- Calculate cumulative frequencies.
- Apply formula M =
th item; N = Total of frequency.
The value (X) corresponding to this in the cumulative frequency will be the median.
c) Continuous Series :
- First calculate model class by inspection or by
- Then apply the following formula -
l1 + ∆1
∆1 + ∆2 where l1 àlower limit of modal class
∆1 àf1 – f0
∆2 àf1 – f2
f1 àfrequency of modal class
f0 àfrequency of preceding class. f2 àfrequency of succeeding class i àclass interval
